That switch could do anything. More often than not, especially on Chinesium power supplies, having a switch like that indicates a lack of a PFC stage.
1) That switch most likely switches which taps on their transformer are used. Say their supply needs 60V on the secondary, for 120 you need a 2:1 ratio, for 240 you need a 4:1 ratio. They may have 2 2:1 primaries that are either wired in series or parallel based on that switch for both voltage ranges. This may be why your output voltage is half of what it should be. Different cheapo supplies use different methods. 120 uses twice the current as 240 for the same amount of power so things need to be sized/done differently.
2) Like I said earlier, your ATX supply most likely has a PFC stage which takes the AC input and boosts it to ~400VDC before the DC/DC converters turn that back into 5, 12, etc. Because the boost can work on anything less than 400Vpk, the AC input can be pretty much anything (the "universal range" is 90-264VAC).
3) The risks? Fire, death, bodily injury, you know, the usual things. If everything in the supply is sized for the 120V currents, then it probably just won't work well, but copper is expensive and corners can be cut. If you see an agency label on it then they test all these things and you'll be fine either way.